Problem 79 - Chemistry: The Central Science

Question

(a) Consider the lattice energies for the following compounds: \(\mathrm{BeH}_{2}, 3205 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{MgH}_{2}, 2791 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{CaH}_{2}, 2410 \mathrm{~kJ} / \mathrm{mol} ;\) \(\mathrm{SrH}_{2}, 2250 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{BaH}_{2}, 2121 \mathrm{~kJ} / \mathrm{mol}\). Plot lattice energy versus cation radius for these compounds. If you draw a line through your points, is the slope negative or positive? Explain. (b) The lattice energy of \(\mathrm{ZnH}_{2}\) is \(2870 \mathrm{~kJ} / \mathrm{mol}\). Based on the data given in part (a), the radius of the \(\mathrm{Zn}^{2+}\) ion is expected to be closest to that of which group \(2 \mathrm{~A}\) element?

Step-by-Step Solution

Answer

The slope of the line in the scatter plot of lattice energy versus cation radius is negative, which indicates that lattice energy decreases as the cation radius increases. This is expected because lattice energy depends on electrostatic interactions between ions, and larger cations have weaker interactions due to their greater distance from the anion. Based on the lattice energy of \(\mathrm{ZnH}_{2}\) (\(2870 \mathrm{~kJ} / \mathrm{mol}\)), the radius of the \(\mathrm{Zn}^{2+}\) ion is estimated to be closest to the radius of the \(\mathrm{Mg}^{2+}\) ion among the given Group \(2 \mathrm{~A}\) elements.

Step 1: Gather ion radius data

To plot lattice energy versus cation radius, you first need the ionic radii of the cations for the compounds you are given (\(\mathrm{Be}^{2+}, \mathrm{Mg}^{2+}, \mathrm{Ca}^{2+}, \mathrm{Sr}^{2+}, \mathrm{Ba}^{2+}\)). You can find these values in a reference book or online.

Step 2: Create a scatter plot

Next, make a scatter plot with cation radius on the x-axis, and lattice energies on the y-axis. Plot the points for the five given hydrides, using the lattice energies from the exercise, and the cation radii you found in step 1.

Step 3: Calculate the slope

Draw a best-fit line through the points in your scatter plot. Determine the slope of this line. If the slope is positive, it means lattice energy increases as cation radius increases; if the slope is negative, it means lattice energy decreases as cation radius increases.

Step 4: Explain the trend

Based on the slope of your line, explain why the lattice energy changes as the cation radius changes. Consider the fact that lattice energy depends on the electrostatic interactions between ions, which in turn depend on the distance between the ions (cation radius).

Step 5: Estimate Zn^2+ ion radius

From the given lattice energy of \(\mathrm{ZnH}_{2}\) (\(2870 \mathrm{~kJ} / \mathrm{mol}\)), estimate the \(\mathrm{Zn}^{2+}\) ion radius by comparing this value to the lattice energies and ion radii in the data you previously used to construct your plot. Find the point on the best-fit line that has a lattice energy closest to \(2870 \mathrm{~kJ} / \mathrm{mol}\), and identify the ion radius corresponding to this point. Based on this estimate, determine which Group \(2 \mathrm{~A}\) element's ion radius is closest to the radius of \(\mathrm{Zn}^{2+}\) ion.

Key Concepts

Cation Radius

The cation radius is a crucial concept when studying lattice energy. Cation radius refers to the size of a positively charged ion, or cation.
As cations form by losing electrons, the radius is generally smaller than that of its neutral atom.
Cation radius influences several properties of ionic compounds:

• It affects the strength of electrostatic interactions within a crystal lattice.
• A smaller cation radius usually results in stronger attractions between ions.

In the original exercise,
you are asked to examine how the lattice energies for various metal hydrides correlate with their respective cation radii. Metals like Be, Mg, Ca, Sr, and Ba are chosen due to their different cation sizes,
which affect how tightly the ions pack together in a crystal.

Electrostatic Interactions

Electrostatic interactions are the forces between charged particles,
and they play a significant role in determining the lattice energy of ionic compounds.
Lattice energy is the energy required to separate one mole of a solid ionic compound into gaseous ions.

• Stronger electrostatic attractions increase the lattice energy.
• The strength of these interactions is inversely related to the distance between the ions, or the cation and anion radii.

In predicting lattice energies,
it's important to consider how changes in cation radius affect the overall structure of the ion lattice.
Smaller cations lead to closer distances between ions, thus stronger interactions and higher lattice energy.

Ionic Radii

The ionic radii refer to the size of an ion, affecting how ions fit together in a crystal lattice.
This directly influences the electrostatic interactions, and thus the compound’s lattice energy.

• Ionic radii are usually measured in picometers or angstroms.
• Differences in ionic radii can lead to varying levels of attraction or repulsion within the lattice.

Understanding ionic radii helps determine the stability of the ionic compound and its lattice energy. When you calculate and interpret lattice energies like those in the original exercise,
comparing the ionic radii allows you to predict trends and make estimations,
as seen with figuring out the radius of Zn^{2+} ion by examining its lattice energy in relation to known values.

Group 2A Elements

Group 2A elements in the periodic table are known as the alkaline earth metals,
which include beryllium, magnesium, calcium, strontium, and barium.

• They form divalent cations (ions with a +2 charge) when they lose two electrons.
• Their cation radii increase as you move down the group.

These elements are significant in studies of lattice energy because their increasing cation radii generally lead to decreased lattice energies.
When analyzing the lattice energies for various hydrides of Group 2A elements,
you see a clear trend; the lattice energy decreases from BeH₂ to BaH₂.
This illustrates how changes in the size of the metal cations affect the interionic forces and the resulting lattice energies.
In exercises involving estimating unknown ionic radii,
such as predicting the radius of Zn^{2+}, it's useful to use these Group 2A elements as a benchmark due to their predictable behavior and trends in their properties.